Ideal structure and pure infiniteness of inverse semigroup crossed products

[摘要] We give efficient conditions under which a C∗\mathrm{C}^*C∗-subalgebra A⊆BA\subseteq BA⊆B separates ideals in a C∗\mathrm{C}^*C∗-algebra BBB, and BBB is purely infinite if every positive element in AAA is properly infinite in BBB. We specialise to the case when BBB is a crossed product for an inverse semigroup action by Hilbert bimodules or a section C∗\mathrm{C}^*C∗-algebra of a Fell bundle over an étale, possibly non-Hausdorff, groupoid. Then our theory works provided BBB is the recently introduced essential crossed product and the action is essentially exact and residually aperiodic or residually topologically free. These last notions are developed in the article.

[学科分类] 神经科学

[关键词] Crossed product;inverse semigroup;Fell bundle;groupoid;purely infinite;primitive ideal space